A018922 Define the generalized Pisot sequence T(a(0),a(1)) by: a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n). This is T(8,16).

8, 16, 31, 60, 116, 224, 432, 833, 1606, 3096, 5968, 11504, 22175, 42744, 82392, 158816, 306128, 590081, 1137418, 2192444, 4226072, 8146016, 15701951, 30266484, 58340524, 112454976, 216763936, 417825921, 805385358, 1552430192, 2992405408, 5768046880

Offset: 0

Author

R. K. Guy

Keywords

• nonn
• easy

Comments

Not to be confused with the Pisot T(8,16), which is essentially A000079. - R. J. Mathar, Feb 13 2016

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
• Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,-1).
• Index entries for Pisot sequences

Crossrefs

Cf. A107066.

Programs

Magma

Tiv:=[8,16]; [n le 2 select Tiv[n] else Ceiling(Self(n-1)^2/Self(n-2))-1: n in [1..40]]; // Bruno Berselli, Feb 17 2016

Mathematica

Drop[CoefficientList[Series[1/(1 - 2*z + z^5), {z, 0, 100}], z], 3] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
RecurrenceTable[{a[1] == 8, a[2] == 16, a[n] == Ceiling[a[n-1]^2/a[n-2]] - 1}, a, {n, 40}] (* Bruno Berselli, Feb 17 2016 *)
LinearRecurrence[{2,0,0,0,-1},{8,16,31,60,116},40] (* Harvey P. Dale, Sep 21 2024 *)

PARI

T(a0, a1, maxn) = a=vector(maxn); a[1]=a0; a[2]=a1; for(n=3, maxn, a[n]=ceil(a[n-1]^2/a[n-2])-1); a
T(8, 16, 40) \\ Colin Barker, Feb 14 2016

Formula

a(n) = 2*a(n-1) - a(n-5).

a(n) = A107066(n+3). - Vladimir Joseph Stephan Orlovsky, Jul 08 2011

O.g.f: -(-8+x^2+2*x^3+4*x^4)/((x-1)*(x^4+x^3+x^2+x-1)) = (1/3)/(x-1)+(1/3)*(-13*x^3-20*x^2-24*x-25)/(x^4+x^3+x^2+x-1) . - R. J. Mathar, Dec 02 2007